How do I make the speed of an animated object constant instead of accelerating?
What you're looking at is the default interpolation (that is, the rate things move between keyframes).  From your description, it is likely that your default interpolation is set to Bezier, which looks like below:



Get into the Graph Editor, select the channel you want to change (the names next to the eye in the above picture).  

Then you want to go to the Menu >> Channel >> Interpolation >> Constant.  It'll move constantly after that.